Lee-Yang–inspired functional with a density-dependent neutron-neutron scattering length

Inspired by the low-density Lee-Yang expansion for the energy of a dilute Fermi gas of density $\rho$ and momentum $k_F$, we introduce here a Skyrme-type functional that contains only $s$-wave terms and provides, at the mean-field level, (i) a satisfactory equation of state for neutron matter from extremely low densities up to densities close to the equilibrium point, and (ii) a good-quality equation of state for symmetric matter at density scales around the saturation point. This is achieved by using a density-dependent neutron-neutron scattering length $a\left(\rho \right)$ which satisfies the low-density limit (for Fermi momenta going to zero) and has a density dependence tuned in such a way that the low-density constraint $|a\left(\rho \right)k_F|\le 1$ is satisfied at all density scales.

Figure 1

Neutron-neutron $s$-wave scattering length as a function of the density (a) and of the Fermi momentum (b). The green region contains all the possible cases between $|k_{F}a\left(k_F\right)|=0.5$ (red dashed line) and $|k_{F}a\left(k_F\right)|=1$ (black solid line).

Figure 2

Density dependence of the parameters $x_0$ (a) and $x_3$ (b). The green region contains all the possible cases between $|k_{F}a\left(k_F\right)|=0.5$ (red dashed line) and $|k_{F}a\left(k_F\right)|=1$ (black solid line).

Figure 3

EOS of symmetric matter obtained within a $t_0-t_3$ model, with the parameters adjusted in Ref. [13] (red dashed line) and within a $t_0-t_3-t_1$ model, with the parameters adjusted in this work (blue circles). For comparison, the SLy5-mean-field EOS is plotted (black solid line).

Figure 4

EOS of neutron matter as a function of the density computed in the mean-field approximation with SLy5 (black triangels), SkP (cyan circles), and SIII (magenta squares). The EOSs obtained with the first two terms of the Lee-Yang expansion by using the constant value $a=-18.9$ fm (blue squares) and a density-dependent scattering length are also shown. For the latter case, the black solid line represents the EOS obtained with the low-density constraint $|k_{F}a\left(k_F\right)|=1$ whereas the red dashed curve represents the EOS obtained by imposing $|k_{F}a\left(k_F\right)|=0.5$. The green area contains the intermediate cases. The black dotted line represents the EOS obtained by imposing $|k_{F}a\left(k_F\right)|=1$ and using an effective range of $-4.5$ fm.

Figure 5

Energy of neutron matter divided by the energy of a free Fermi gas as a function of $|a{k}_F|$, with $a=-18.9\mathrm{fm}$. The black solid and the blue dotted curves represent the SLy5-mean field and the LY (with only the first two terms and $a=-18.9\mathrm{fm}$) EOSs, respectively. The red dashed and the green dot-dashed curves illustrate the EOSs obtained with a density-dependent scattering length with $|a\left(k_F\right)k_f|=1$, with only the first two terms and with the full expression of Eq. (1) ($r_s=-4.5\mathrm{fm}$), respectively. The ab initio quantum Monte Carlo (QMC) results with only $s$ wave and with the full AV4 interaction taken from Refs. [24, 25] are shown respectively with purple squares and blue triangles. The AFDMC ab initio results are taken from Ref. [26].